This analysis examines the stimulus-response characateristics of the SPARS.

Unlike Trial A, where participants were exposed to a prescribed range of stimulus intensities (1 to 4J, at 0.25J intervals), in Trial B, all participants were exposed to 9 stimulus intensities (0.25J interval), but the range of stimuli intensities were calibrated against the sensitivity of each participant. For example, a more sensitive participant may have been exposed to 9 stimuli from 1.75J to 3.75J, another participant may be exposed to stimuli from 2.5J to 4.5J.

This design makes performing group-level analyses difficult (the extremes will have fewer observations), and so we transformed the exposure intensities into an relative scale by ranking (1 to 9) an ordered list of 9 stimulus intensities for each participant. This brought everyone onto the same 1 to 9 scale.

For transparency, we have performed exploratory plots using the raw stimulus intensity data and the relative intensity data. However, the regression analysis was performed using the relative intenisty data only.

Import and inspect data

# Import
data <- read_rds('./data-cleaned/SPARS_B.rds')

# Inspect
glimpse(data)

## Observations: 2,256
## Variables: 9
## $ PID             <chr> "ID06", "ID06", "ID06", "ID06", "ID06", "ID06"...
## $ scale           <chr> "SPARS", "SPARS", "SPARS", "SPARS", "SPARS", "...
## $ block_number    <int> 2, 2, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 11, 11, 11...
## $ trial_number    <int> 4, 6, 27, 9, 13, 20, 20, 24, 27, 4, 18, 22, 2,...
## $ intensity       <dbl> 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75...
## $ intensity_char  <chr> "1.75", "1.75", "1.75", "1.75", "1.75", "1.75"...
## $ intensity_rank  <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ rating          <int> -49, 2, -6, 3, -20, -2, -31, 2, -5, -8, -23, 1...
## $ rating_positive <dbl> 1, 52, 44, 53, 30, 48, 19, 52, 45, 42, 27, 64,...

Clean and process data

We performed a basic clean-up of the data, and then calculated Tukey trimean at each stimulus intensity for each participant (participant average), and finally the median of the trimeans at each stimulus intensity across participants (group average).

############################################################
#                                                          #
#                          Clean                           #
#                                                          #
############################################################
data %<>%
    # Rename block_number
    rename(block = block_number) %>% 
    # Select SPARS scale
    filter(scale == 'SPARS') %>% 
    ungroup() %>%
    arrange(PID)

############################################################
#                                                          #
#                Calculate 'Tukey trimean'                 #
#                                                          #
############################################################
# Define tri.mean function
tri.mean <- function(x) {
  # Calculate quantiles
  q1 <- quantile(x, probs = 0.25, na.rm = TRUE)[[1]]
  q2 <- median(x, na.rm = TRUE)
  q3 <- quantile(x, probs = 0.75, na.rm = TRUE)[[1]]
  # Calculate trimean
  tm <- (q2 + ((q1 + q3) / 2)) / 2
  # Convert to integer
  tm <- as.integer(round(tm))
  return(tm)
}

# Calculate the participant average based on 'raw' intensity
data_tm <- data %>% 
  group_by(PID, intensity) %>%
  summarise(tri_mean = tri.mean(rating)) %>% 
  ungroup()

# Calculate the group average based on 'raw' intensity
data_group <- data_tm %>%
  group_by(intensity) %>%
  summarise(median = median(tri_mean)) %>%
  ungroup() 

# Calculate the participant average based on 'relative' intensity
data_tmR <- data %>% 
  group_by(PID, intensity_rank) %>%
  summarise(tri_mean = tri.mean(rating)) %>%
  ungroup()

# Calculate the group average based on 'relative' intensity
data_groupR <- data_tmR %>%
  group_by(intensity_rank) %>%
  summarise(median = median(tri_mean)) %>%
  ungroup()

Stimulus exposure ranges

knitr::kable(data %>% 
                 group_by(PID) %>% 
                 summarise(`Minimum stimulus intensity` = min(intensity),
                           `Maximum stimulus intensity` = max(intensity)),
             caption = 'Range of stimulus intensities covered in each participant')

Range of stimulus intensities covered in each participant
PID	Minimum stimulus intensity	Maximum stimulus intensity
ID01	2.25	4.25
ID02	2.25	4.25
ID03	2.50	4.50
ID04	2.50	4.50
ID05	2.50	4.50
ID06	1.75	3.75
ID07	2.25	4.25

Exploratory plots

Group-level stimulus response curve

# Plot (y.axis = raw stimulus intensity)
data_tm %>%
  ggplot(data = .) +
  aes(x = intensity,
      y = tri_mean) +
  geom_point(position = position_jitter(width = 0.05)) +
  geom_smooth(method = 'loess',
              se = FALSE,
              colour = '#666666', 
              size = 0.6) +
  geom_point(data = data_group,
             aes(y = median),
             shape = 21,
             size = 4,
             fill = '#D55E00') +
  labs(title = 'Group-level stimulus-response plots (raw intensity)',
       subtitle = 'Black circles: participant-level Tukey trimeans | Orange circles: group-level median | Grey line: loess curve',
       x = 'Stimulus intensity (J)',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  scale_x_continuous(breaks = seq(from = 1, to = 4, by = 0.5))

# Plot (y.axis = relative stimulus intensity)
data_tmR %>%
  ggplot(data = .) +
  aes(x = intensity_rank,
      y = tri_mean) +
  geom_point(position = position_jitter(width = 0.05)) +
  geom_smooth(method = 'loess',
              se = FALSE,
              colour = '#666666', 
              size = 0.6) +
  geom_point(data = data_groupR,
             aes(y = median),
             shape = 21,
             size = 4,
             fill = '#D55E00') +
  labs(title = 'Group-level stimulus-response plots (relative intensity)',
       subtitle = 'Black circles: participant-level Tukey trimeans | Orange circles: group-level median | Grey line: loess curve\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
       x = 'Relative stimulus intensity',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  scale_x_continuous(breaks = seq(from = 1, to = 9, by = 1))

Participant-level stimulus response curves

All trials

# Plot (y.axis = raw stimulus intensity)
data %>%
  ggplot(data = .) +
  aes(x = intensity,
      y = rating) +
  geom_point() +
  geom_smooth(method = 'loess',
              se = FALSE,
              colour = '#666666',
              size = 0.6) +
  geom_point(data = data_tm,
             aes(y = tri_mean),
             shape = 21,
             size = 3,
             fill = '#D55E00') +
  labs(title = 'Participant-level stimulus-response plot (raw intensity)',
       subtitle = 'Black circles: individual experimental blocks | Orange circles: Tukey trimean | Grey line: loess curve',
       x = 'Stimulus intensity (J)',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  facet_wrap(~ PID, ncol = 4)

# Plot (y.axis = rank stimulus intensity)
data %>%
  ggplot(data = .) +
  aes(x = intensity_rank,
      y = rating) +
  geom_point() +
  geom_smooth(method = 'loess',
              se = FALSE,
              colour = '#666666',
              size = 0.6) +
  geom_point(data = data_tmR,
             aes(y = tri_mean),
             shape = 21,
             size = 3,
             fill = '#D55E00') +
  labs(title = 'Participant-level stimulus-response plot (relative intensity)',
       subtitle = 'Black circles: individual experimental blocks | Orange circles: Tukey trimean | Grey line: loess curve\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
       x = 'Relative stimulus intensity',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  facet_wrap(~ PID, ncol = 4)

Trials by experimental block

# Process data (raw stimulus intensity)
data_block <- data %>%
  # Rename blocks
  #mutate(block = sprintf('Block: %s (order: %i)', block, block_order)) %>%
  # Nest by PID
  group_by(PID) %>%
  nest() %>%
  # Generate plots
  mutate(plots = map2(.x = data,
                      .y = unique(PID),
                      ~ ggplot(data = .x) +
                        aes(x = intensity,
                            y = rating) +
                        geom_point() +
                        geom_smooth(method = 'loess',
                                    se = FALSE,
                                    colour = '#666666',
                                    size = 0.6) +
                        labs(title = paste(.y, ': Participant-level stimulus-response plots conditioned on experimental block (raw intensity'),
                             subtitle = 'Black circles: individual data points | Grey line: loess curve',
                             x = 'Stimulus intensity (J)',
                             y = 'SPARS rating [-50 to 50]') +
                        scale_y_continuous(limits = c(-50, 50)) +
                        facet_wrap(~ block, ncol = 2)))

# Print plots
walk(.x = data_block$plots, ~ print(.x))

# Process data (relative stimulus intensity)
data_blockR <- data %>%
  # Rename blocks
  #mutate(block = sprintf('Block: %s (order: %i)', block, block_order)) %>%
  # Nest by PID
  group_by(PID) %>%
  nest() %>%
  # Generate plots
  mutate(plots = map2(.x = data,
                      .y = unique(PID),
                      ~ ggplot(data = .x) +
                        aes(x = intensity_rank,
                            y = rating) +
                        geom_point() +
                        geom_smooth(method = 'loess',
                                    se = FALSE,
                                    colour = '#666666',
                                    size = 0.6) +
                        labs(title = paste(.y, ': Participant-level stimulus-response plots conditioned on experimental block (relative intensity'),
                             subtitle = 'Black circles: individual data points | Grey line: loess curve\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
                             x = 'Relative stimulus intensity',
                             y = 'SPARS rating [-50 to 50]') +
                        scale_y_continuous(limits = c(-50, 50)) +
                        facet_wrap(~ block, ncol = 2)))

# Print plots
walk(.x = data_blockR$plots, ~ print(.x))

Linear mixed model regression

To allow for a curvilinear relationship between stimulus intensity and rating, we modelled the data using polynomial regression, with 1^st (linear), 2^nd (quadratic), and 3^rd (cubic) order orthogonal polynomials. For each polynomial expression, we modelled the random effects as random intercept only, and as random intercept and slope.

The random intercept only and random intercept and slope models were compared using the logliklihood test, and the better model taken foward. Diagnostics were run on the final model only, and we examined level 1 residuals (conditional / fixed effects), and level 2 residuals (random effects) and influence points ¹.

1st-order (linear) polynomial

# Intercept only
lmm1 <- lmer(tri_mean ~ intensity_rank + (1 | PID),
             data = data_tmR,
             REML = TRUE)

# Intercept and slope
lmm1b <- lmer(tri_mean ~ intensity_rank + (intensity_rank | PID),
              data = data_tmR,
              REML = TRUE)

# Better model?
anova(lmm1, lmm1b)

## Data: data_tmR
## Models:
## lmm1: tri_mean ~ intensity_rank + (1 | PID)
## lmm1b: tri_mean ~ intensity_rank + (intensity_rank | PID)
##       Df    AIC    BIC  logLik deviance Chisq Chi Df Pr(>Chisq)    
## lmm1   4 495.89 504.46 -243.94   487.89                            
## lmm1b  6 475.23 488.08 -231.61   463.23 24.66      2  4.418e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Anova of better model
Anova(lmm1b,
      type = 2,
      test.statistic = 'F')

## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
## 
## Response: tri_mean
##                     F Df Df.res   Pr(>F)   
## intensity_rank 28.612  1      6 0.001746 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Print better model
sjt.lmer(lmm1b,
         show.header = TRUE,
         string.dv = "Response", 
         string.pred = "Coefficients",
         depvar.labels = '',
         pred.labels = 'intensity_rank',
         string.est = 'Estimate',
         string.ci = '95% CI',
         string.p = 'p-value',
         show.icc = FALSE,
         show.r2 = FALSE)

Coefficients	Response
Coefficients
		Estimate	95% CI	p-value
Fixed Parts
(Intercept)		-38.87	-56.10 – -21.65	.004
intensity_rank		6.01	3.81 – 8.21	.002
Random Parts
σ²		58.333
τ_{00, PID}		509.896
ρ₀₁		-0.958
N_PID		7
Observations		63

2nd-order (quadratic) polynomial

# Intercept only
lmm2 <- lmer(tri_mean ~ poly(intensity_rank, 2) + (1 | PID),
             data = data_tmR,
             REML = TRUE)

# Intercept and slope
lmm2b <- lmer(tri_mean ~ poly(intensity_rank, 2) + (intensity_rank | PID),
              data = data_tmR,
              REML = TRUE)

# Better model?
anova(lmm2, lmm2b)

## Data: data_tmR
## Models:
## lmm2: tri_mean ~ poly(intensity_rank, 2) + (1 | PID)
## lmm2b: tri_mean ~ poly(intensity_rank, 2) + (intensity_rank | PID)
##       Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
## lmm2   5 495.50 506.22 -242.75   485.50                             
## lmm2b  7 472.71 487.71 -229.35   458.71 26.793      2  1.521e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Anova for better model
Anova(lmm2b,
      type = 2,
      test.statistic = 'F')

## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
## 
## Response: tri_mean
##                             F Df Df.res    Pr(>F)    
## poly(intensity_rank, 2) 15.85  2 13.527 0.0002848 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Print better model
sjt.lmer(lmm2b,
         show.header = TRUE,
         string.dv = "Response", 
         string.pred = "Coefficients",
         depvar.labels = '',
         pred.labels = 'intensity_rank',
         string.est = 'Estimate',
         string.ci = '95% CI',
         string.p = 'p-value',
         show.icc = FALSE,
         show.r2 = FALSE)

Coefficients	Response
Coefficients
		Estimate	95% CI	p-value
Fixed Parts
(Intercept)		-8.83	-16.47 – -1.18	.064
poly(intensity_rank, 2)1		123.16	78.03 – 168.29	.002
poly(intensity_rank, 2)2		15.87	1.42 – 30.31	.075
Random Parts
σ²		54.304
τ_{00, PID}		512.022
ρ₀₁		-0.957
N_PID		7
Observations		63

3rd-order (cubic) polynomial

# Intercept only
lmm3 <- lmer(tri_mean ~ poly(intensity_rank, 3) + (1 | PID),
             data = data_tmR,
             REML = TRUE)

# Intercept and slope
lmm3b <- lmer(tri_mean ~ poly(intensity_rank, 3) + (intensity_rank | PID),
              data = data_tmR,
              REML = TRUE)

# Better model?
anova(lmm3, lmm3b)

## Data: data_tmR
## Models:
## lmm3: tri_mean ~ poly(intensity_rank, 3) + (1 | PID)
## lmm3b: tri_mean ~ poly(intensity_rank, 3) + (intensity_rank | PID)
##       Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
## lmm3   6 497.33 510.19 -242.66   485.33                             
## lmm3b  8 474.37 491.52 -229.19   458.37 26.956      2  1.402e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Anova for better model
Anova(lmm3b,
      type = 2,
      test.statistic = 'F')

## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
## 
## Response: tri_mean
##                              F Df Df.res    Pr(>F)    
## poly(intensity_rank, 3) 10.617  3 21.094 0.0001842 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Print better model
sjt.lmer(lmm3b,
         show.header = TRUE,
         string.dv = "Response", 
         string.pred = "Coefficients",
         depvar.labels = '',
         pred.labels = 'intensity_rank',
         string.est = 'Estimate',
         string.ci = '95% CI',
         string.p = 'p-value',
         show.icc = FALSE,
         show.r2 = FALSE)

Coefficients	Response
Coefficients
		Estimate	95% CI	p-value
Fixed Parts
(Intercept)		-8.83	-16.47 – -1.18	.064
poly(intensity_rank, 3)1		123.16	78.03 – 168.29	.002
poly(intensity_rank, 3)2		15.87	1.32 – 30.41	.076
poly(intensity_rank, 3)3		-4.22	-18.76 – 10.33	.590
Random Parts
σ²		55.081
τ_{00, PID}		511.612
ρ₀₁		-0.957
N_PID		7
Observations		63

Compare models

knitr::kable(broom::tidy(anova(lmm1b, lmm2b, lmm3b)),
             caption = 'Linear model vs quadratic model and cubic model')

Linear model vs quadratic model and cubic model
term	df	AIC	BIC	logLik	deviance	statistic	Chi.Df	p.value
lmm1b	6	475.2254	488.0842	-231.6127	463.2254	NA	NA	NA
lmm2b	7	472.7085	487.7104	-229.3542	458.7085	4.5169285	1	0.0335610
lmm3b	8	474.3731	491.5182	-229.1866	458.3731	0.3353413	1	0.5625307

PLot the model

predicted <- ggeffects::ggpredict(model = lmm2b,
                                  terms = 'intensity_rank',
                                  ci.lvl = 0.95) 
ggplot() +
    geom_ribbon(data = predicted,
                aes(x = x,
                    ymin = conf.low,
                    ymax = conf.high),
                fill = '#cccccc') + 
    geom_line(data = predicted,
              aes(x = x,
                  y = predicted)) +
    geom_point(data = predicted,
              aes(x = x,
                  y = predicted)) +
    geom_point(data = data_groupR,
               aes(x = intensity_rank,
                   y = median),
               shape = 21,
               size = 4,
               fill = '#D55E00') +
  labs(title = 'Quadratic model (95% CI): Predicted values vs stimulus intensity_rank',
       subtitle = 'Black circles/line: predicted values | Orange circles: group-level median\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
       x = 'Relative stimulus intensity',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  scale_x_continuous(breaks = seq(from = 1, to = 9, by = 1))

The quadratic model has the best fit. We performed diagnostics on this model to confirm that the model was properly specified.

Diagnostics on the cubic model

Generate residuals

# Level 1 residuals
## Standardized
lmm_resid1 <- HLMresid(lmm2b,
                       level = 1,
                       type = 'LS',
                       standardize = TRUE)

# Semi-standardized residuals (used for assessing homoscedasticity)
lmm_ssresid1 <- HLMresid(lmm2b,
                         level = 1,
                         type = 'LS',
                         standardize = 'semi')

# Level 2 residuals
## Standardized
lmm_resid2 <- HLMresid(lmm2b,
                       level = 'PID',
                       type = 'EB')

Level 1 residuals: linearity

The relationship between predictor(s) and outcome for a linear model should be linear. This relationship can be observed by plotting the level 1 standardized residuals against the predictors. The scatter of residuals should show no pattern, and be centered around 0.

# Standardized residuals vs intensity_rank
ggplot(data = lmm_resid1) +
    aes(x = `poly(intensity_rank, 2)`[, 1],
        y = std.resid) +
    geom_point() +
    geom_smooth(method = 'lm') +
    geom_hline(yintercept = 0) +
    geom_hline(yintercept = -2,
               linetype = 2) +
    geom_hline(yintercept = 2,
               linetype = 2) +
    labs(title = 'Quadratic model: Level 1 residuals vs intensity_rank',
         subtitle = 'Assess linearity of the intensity_rank term | Blue line: linear regression line',
         caption = 'The regression line should be centered on 0\n~95% of points should be betwen -2 and +2',
         y = 'Standardized residuals',
         x = 'Stimulus intensity_rank')

# Standardized residuals vs intensity^2
ggplot(data = lmm_resid1) +
    aes(x = `poly(intensity_rank, 2)`[, 2],
        y = std.resid) +
    geom_point() +
    geom_smooth(method = 'lm') +
    geom_hline(yintercept = 0) +
    geom_hline(yintercept = -2,
               linetype = 2) +
    geom_hline(yintercept = 2,
               linetype = 2) +
    labs(title = expression(paste('Quadratic model: Level 1 residuals vs ', intensity^2)),
         subtitle = 'Assess linearity of the intensity_rank^2 term | Blue line: linear regression line\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
         caption = 'The regression line should be centered on 0\n~95% of points should be betwen -2 and +2',
         y = 'Standardized residuals',
         x = expression(Relative~stimulus~intensity^2))

Based on the plot of the linear and quadratic terms’ residuals, we accept that the condition of linearity for the quadratic model.

Level 1 residuals: homoscedasticity

The variance of residuals should be constant across the range of the predictor(s). This relationship can be observed by plotting the level 1 semi-standardized residuals against the predictors. Like the assessment of linearity, the residuals should be centered on 0, and show no pattern in the scatter of points.

# Standardized residuals vs intensity_rank
ggplot(data = lmm_ssresid1) +
    aes(x = `poly(intensity_rank, 2)`[ ,1],
        y = semi.std.resid) +
    geom_point() +
    geom_hline(yintercept = 0) +
    labs(title = 'Quadratic model: Level 1 residuals vs intensity_rank',
         subtitle = 'Assess homoscedasticity for the intensity_rank term',
         y = 'Semi-standardized residuals',
         x = 'Relative stimulus intensity')

# Standardized residuals vs intensity^2
ggplot(data = lmm_ssresid1) +
    aes(x = `poly(intensity_rank, 2)`[, 2],
        y = semi.std.resid) +
    geom_point() +
    geom_smooth(method = 'lm') +
    geom_hline(yintercept = 0) +
    labs(title = expression(paste('Quadratic model: Level 1 residuals vs ', intensity^2)),
         subtitle = 'Assess homoscedasticity for the intensity_rank^2 term | Blue line: linear regression line\nRelative intensity was calculated using the rank of the ordered (ascending) stimulus intensities each participant was exposed to.',
         y = 'Semi-standardized residuals',
         x = expression(Relative~stimulus~intensity^2))

There is no obvious pattern to the scatter of residuals across any of the fixed effect terms. So we accept that the residuals are homoscedastic in the quadratic model.

Level 1 residuals: residual distribution

Residuals should be normally distributed. There are various methods of examining the distribution, and we have chosen the QQ-plot method, which plots the quantiles of the standardized residuals against a theoretical (Gaussian) quantile distribution. Points should line on the line of identity of the two sets of quantiles follow the same distribution.

# Standardized residuals vs intensity_rank
ggplot_qqnorm(x = lmm_resid1$std.resid, 
              line = "rlm") +
    labs(title = 'Quadratic model: QQ-plot of level 1 residuals',
         subtitle = 'Assessing whether residuals follow a normal distribution',
         x = 'Theoretical quantiles',
         y = 'Standardized residuals')

There is minor deviation at the extremes (possibly a thin left tail), but on the whole, we are satisfied that the quadratic model fits the assumption of normally distributed residuals.

Level 2 residuals: residual distribution

Level 2 residuals can be used to identify predictors that should be included in the model, but since we are only assessing the effect of stimulus strength on SPARS rating, we have only assessed whether the level 2 residuals (intercept and slope) meet the assumption of being normally distributed (assessed using QQ-plots).

# Generate QQplots 
qq1 <- ggplot_qqnorm(x = lmm_resid2$`(Intercept)`, 
              line = "rlm") +
    labs(title = 'Quadratic model: QQ-plot of level 2 residuals (Intercept)',
         subtitle = 'Assessing whether residuals follow a normal distribution',
         x = 'Theoretical quantiles',
         y = 'Residuals') 

qq2 <- ggplot_qqnorm(x = lmm_resid2$intensity_rank, 
              line = "rlm") +
    labs(title = 'Quadratic model: QQ-plot of level 2 residuals (slope: intensity_rank)',
         subtitle = 'Assessing whether residuals follow a normal distribution',
         x = 'Theoretical quantiles',
         y = 'Residuals') 

# Plot
qq1 + qq2

Although the data are sparse, we are satisfied that the level 2 residuals for the intercept and the slope of the quadratic model fit the assumption of being normally distributed.

influence points

We assessed three aspects of influence (data that significantly model coefficients):

The variance component (random effects) was assesed using the relative variance change metric, which calculates the impact of deleting observational units of the variance of the residuals, random intercept, random slope, and covariance of the random slope and random intercept.
Leverage was used to assess fitted values. The assessment involves assessing the rate of change in the predicted response with respect to the observed response.
Cook’s Distance was used to assess the influence of fixed effects. The metric measures the distance between the fixed effects estimates obtained from the full model to that obtained from the reduced data (observations removed).

In all cases, we treated the individual (indicated using PID) as the unit of observation, and we used internal scaling to set the diagnostic cutoffs for each metric. The cutoffs were determined as: \(3^{rd}~Quartile + (3 \cdot IQR)\).

# Prepare relative variance change (RCV)
influence_rvc <- rvc(lmm1b, 
                     group = 'PID')

# Prepare Cook's distance
influence_cooks <- cooks.distance(lmm1b, 
                                  group = 'PID')

# Prepare leverage 
## (Assessed at the level of PID, and not the individual observation)
influence_leverage <- leverage(lmm1b,
                               level = 'PID')

Random effects

Estimation of the variance component was undertaken by calculating relative variance change (RCV). RVC is close to zero when deletion of observational units from the model does not have a large infuence on the variance component.

# Plot
dotplot_diag(x = influence_rvc[ , 1], 
             cutoff = 'internal',
             name = 'rvc') + 
    labs(title = 'Relative variance change for the residual variance',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Relative variance change',
         x = 'Participant ID')

dotplot_diag(x = influence_rvc[ , 2], 
             cutoff = 'internal',
             name = 'rvc') + 
    labs(title = 'Relative variance change for the random intercept variance',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Relative variance change',
         x = 'Participant ID')

dotplot_diag(x = influence_rvc[ , 3], 
             cutoff = 'internal',
             name = 'rvc') + 
    labs(title = 'Relative variance change for the random slope variance',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Relative variance change',
         x = 'Participant ID')

dotplot_diag(x = influence_rvc[ , 4], 
             cutoff = 'internal',
             name = 'rvc') + 
    labs(title = 'Relative variance change for the random slope and intercept covariance',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Relative variance change',
         x = 'Participant ID')

One value (PID11) is below the cutoff for the relative variance change for random slope and intercept covariance. The extent of the deviation is minor, and was ignored.

Fitted values

Assessing whether observations are unusual with regard to the fitted values and explanatory variables using leverage. We assessed leverage at two levels: i) fixed effects, and ii) unconfounded (by fixed effects) random effects.

dotplot_diag(x = influence_leverage[, 2], 
             cutoff = "internal",
             name = "leverage") + 
    labs(title = 'Leverage: fixed effects',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Leverage',
         x = 'Participant ID')

dotplot_diag(x = influence_leverage[, 4], 
             cutoff = "internal",
             name = "leverage") + 
    labs(title = 'Leverage: unconfounded random effects',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Leverage',
         x = 'Participant ID')

Fixed effects

Influence points were assessed by calculating Cook’s Distance metrics.

# Plot data
dotplot_diag(x = influence_cooks, 
             cutoff = "internal",
             name = "cooks.distance") + 
    labs(title = 'Influence: Cooks Distance',
         subtitle = 'Cutoffs determined by measures of internal scaling',
         y = 'Cooks Distance',
         x = 'Participant ID')

Based on There are no influential fixed effects.

Summary

The linear is well-specified.

Quantile mixed model regression

# Quantile model with 2.5, 25, 50, 75, and 97.5% quantiles
qmm <- lqmm(fixed = tri_mean ~ poly(intensity_rank, 2),
            random = ~ intensity_rank,
            group = PID,
            data = data_tmR,
            tau = c(0.025, 0.25, 0.5, 0.75, 0.975))

# Summary 
summary(qmm)

## Call: lqmm(fixed = tri_mean ~ poly(intensity_rank, 2), random = ~intensity_rank, 
##     group = PID, tau = c(0.025, 0.25, 0.5, 0.75, 0.975), data = data_tmR)
## 
## tau = 0.025
## 
## Fixed effects:
##                             Value Std. Error lower bound upper bound
## (Intercept)               -69.078     22.134    -113.558     -24.599
## poly(intensity_rank, 2)1  128.368     18.632      90.924     165.811
## poly(intensity_rank, 2)2   11.399     14.729     -18.201      40.998
##                          Pr(>|t|)    
## (Intercept)               0.00302 ** 
## poly(intensity_rank, 2)1 9.84e-09 ***
## poly(intensity_rank, 2)2  0.44272    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## tau = 0.25
## 
## Fixed effects:
##                             Value Std. Error lower bound upper bound
## (Intercept)              -17.4995     8.6741    -34.9308     -0.0682
## poly(intensity_rank, 2)1 121.5351    19.2075     82.9361    160.1341
## poly(intensity_rank, 2)2   5.7801    14.6551    -23.6705     35.2307
##                           Pr(>|t|)    
## (Intercept)                0.04914 *  
## poly(intensity_rank, 2)1 7.305e-08 ***
## poly(intensity_rank, 2)2   0.69499    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## tau = 0.5
## 
## Fixed effects:
##                             Value Std. Error lower bound upper bound
## (Intercept)               -4.4901     5.8081    -16.1619      7.1817
## poly(intensity_rank, 2)1 121.6123    19.9464     81.5285    161.6961
## poly(intensity_rank, 2)2  13.7000    14.7175    -15.8759     43.2759
##                           Pr(>|t|)    
## (Intercept)                 0.4432    
## poly(intensity_rank, 2)1 1.659e-07 ***
## poly(intensity_rank, 2)2    0.3565    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## tau = 0.75
## 
## Fixed effects:
##                             Value Std. Error lower bound upper bound
## (Intercept)                2.4089    12.2317    -22.1717      26.989
## poly(intensity_rank, 2)1 118.7761    22.3803     73.8013     163.751
## poly(intensity_rank, 2)2  17.9548    15.5720    -13.3383      49.248
##                           Pr(>|t|)    
## (Intercept)                 0.8447    
## poly(intensity_rank, 2)1 2.676e-06 ***
## poly(intensity_rank, 2)2    0.2545    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## tau = 0.975
## 
## Fixed effects:
##                            Value Std. Error lower bound upper bound
## (Intercept)               60.212     20.260      19.497     100.926
## poly(intensity_rank, 2)1 117.400     19.658      77.897     156.904
## poly(intensity_rank, 2)2  17.274     14.273     -11.409      45.956
##                           Pr(>|t|)    
## (Intercept)               0.004577 ** 
## poly(intensity_rank, 2)1 2.583e-07 ***
## poly(intensity_rank, 2)2  0.232001    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Null model (likelihood ratio):
## [1] 33.963 (p = 4.218e-08) 70.106 (p = 5.551e-16) 25.873 (p = 2.409e-06)
## [4]  3.623 (p = 1.634e-01) 29.445 (p = 4.037e-07)
## AIC:
## [1] 585.5 (df = 6) 509.7 (df = 6) 522.3 (df = 6) 524.7 (df = 6)
## [5] 562.1 (df = 6)

# Get predicted values
## Level 0 (conditional, note difference to the lmer diagnostics)
quant_predict <- as.data.frame(predict(qmm, level = 0))
names(quant_predict) <- paste0('Q', c(2.5, 25, 50, 75, 97.5))

# Join with 'central_lmm'
data_lqmm <- data_tmR %>%
  bind_cols(quant_predict)

# Trim prediction to upper and lower limits of the scale
data_lqmm %<>%
  mutate_if(is.numeric,
            funs(ifelse(. > 50,
                        yes = 50,
                        no = ifelse(. < -50,
                                    yes = -50,
                                    no = .))))

# Plot
ggplot(data = data_lqmm) +
  aes(x = intensity_rank,
      y = Q50) +
  geom_ribbon(aes(ymin = `Q2.5`,
                  ymax = `Q97.5`),
              fill = '#E69F00') +
  geom_ribbon(aes(ymin = `Q25`,
                  ymax = `Q75`),
              fill = '#56B4E9') +
  geom_point(size = 3,
             shape = 21,
             fill = '#FFFFFF',
             colour = '#000000') +
  geom_hline(yintercept = 0,
             linetype = 2) +
  labs(title = paste('Quantile regression'),
       subtitle = 'Open circles: 50th percentile (median) | Blue band: interquartile range | Orange band: 95% prediction interval',
       x = 'Realtive stimulus intensity',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  scale_x_continuous(breaks = unique(data_lqmm$intensity_rank))

## With original data
ggplot(data = data_lqmm) +
  aes(x = intensity_rank,
      y = Q50) +
  geom_ribbon(aes(ymin = `Q2.5`,
                  ymax = `Q97.5`),
              fill = '#E69F00') +
  geom_ribbon(aes(ymin = `Q25`,
                  ymax = `Q75`),
              fill = '#56B4E9') +
  geom_point(data = data_tmR,
             aes(y = tri_mean),
             position = position_jitter(width = 0.03)) +
  geom_point(size = 3,
             shape = 21,
             fill = '#FFFFFF',
             colour = '#000000') +
  geom_hline(yintercept = 0,
             linetype = 2) +
  labs(title = paste('Quantile regression (with original Tukey trimean data)'),
       subtitle = 'Open circles: 50th percentile (median) | Blue band: interquartile range | Orange band: 95% prediction interval',
       x = 'Stimulus intensity_rank (J)',
       y = 'SPARS rating [-50 to 50]') +
  scale_y_continuous(limits = c(-50, 50)) +
  scale_x_continuous(breaks = unique(data_lqmm$intensity_rank))

The response is consistent across the range of stimulus intensities, but the preduction interval is extremely broad.

Session information

sessionInfo()

## R version 3.4.3 (2017-11-30)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.3
## 
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
## 
## attached base packages:
## [1] methods   stats     graphics  grDevices utils     datasets  base     
## 
## other attached packages:
##  [1] bindrcpp_0.2       car_2.1-6          sjPlot_2.4.1      
##  [4] HLMdiag_0.3.1      lqmm_1.5.3         lme4_1.1-15       
##  [7] Matrix_1.2-12      patchwork_0.0.1    forcats_0.2.0     
## [10] stringr_1.2.0      dplyr_0.7.4        purrr_0.2.4       
## [13] readr_1.1.1        tidyr_0.8.0        tibble_1.4.2      
## [16] ggplot2_2.2.1.9000 tidyverse_1.2.1    magrittr_1.5      
## 
## loaded via a namespace (and not attached):
##   [1] TH.data_1.0-8      minqa_1.2.4        colorspace_1.3-2  
##   [4] modeltools_0.2-21  sjlabelled_1.0.7   rprojroot_1.3-2   
##   [7] estimability_1.2   snakecase_0.8.1    rstudioapi_0.7    
##  [10] glmmTMB_0.2.0      MatrixModels_0.4-1 DT_0.4            
##  [13] mvtnorm_1.0-7      lubridate_1.7.1    coin_1.2-2        
##  [16] xml2_1.2.0         codetools_0.2-15   splines_3.4.3     
##  [19] mnormt_1.5-5       knitr_1.19         sjmisc_2.7.0      
##  [22] effects_4.0-0      bayesplot_1.4.0    jsonlite_1.5      
##  [25] nloptr_1.0.4       ggeffects_0.3.1    pbkrtest_0.4-7    
##  [28] broom_0.4.3        shiny_1.0.5        compiler_3.4.3    
##  [31] httr_1.3.1         sjstats_0.14.1     emmeans_1.1       
##  [34] backports_1.1.2    assertthat_0.2.0   lazyeval_0.2.1    
##  [37] survey_3.33        cli_1.0.0          quantreg_5.34     
##  [40] htmltools_0.3.6    tools_3.4.3        SparseGrid_0.8.2  
##  [43] coda_0.19-1        gtable_0.2.0       glue_1.2.0        
##  [46] reshape2_1.4.3     merTools_0.3.0     Rcpp_0.12.15      
##  [49] carData_3.0-0      cellranger_1.1.0   nlme_3.1-131      
##  [52] psych_1.7.8        lmtest_0.9-35      rvest_0.3.2       
##  [55] mime_0.5           stringdist_0.9.4.6 MASS_7.3-48       
##  [58] zoo_1.8-1          scales_0.5.0.9000  hms_0.4.1         
##  [61] parallel_3.4.3     sandwich_2.4-0     SparseM_1.77      
##  [64] pwr_1.2-1          TMB_1.7.12         yaml_2.1.16       
##  [67] stringi_1.1.6      highr_0.6          blme_1.0-4        
##  [70] rlang_0.1.6        pkgconfig_2.0.1    arm_1.9-3         
##  [73] evaluate_0.10.1    lattice_0.20-35    prediction_0.2.0  
##  [76] bindr_0.1          labeling_0.3       htmlwidgets_1.0   
##  [79] tidyselect_0.2.3   plyr_1.8.4         R6_2.2.2          
##  [82] multcomp_1.4-8     RLRsim_3.1-3       withr_2.1.1.9000  
##  [85] pillar_1.1.0       haven_1.1.1        foreign_0.8-69    
##  [88] mgcv_1.8-23        survival_2.41-3    abind_1.4-5       
##  [91] nnet_7.3-12        modelr_0.1.1       crayon_1.3.4      
##  [94] rmarkdown_1.8      grid_3.4.3         readxl_1.0.0      
##  [97] digest_0.6.15      xtable_1.8-2       httpuv_1.3.5      
## [100] stats4_3.4.3       munsell_0.4.3

Loy A, Hofmann H. HLMdiag: A suite of diagnostics for hierarchical linear models in R. J. Stat. Softw. 2014;56:1–28. Available ↩

SPARS trial B

Stimulus-response characteristics of the SPARS

Peter Kamerman

08 Feb 2018